Dynamic evaluation of a delayed financial system influenced by profitability feedback

This paper presents a comprehensive investigation of the complex dynamical behavior and bifurcation structure of a nonlinear financial system incorporating time-delay feedback and average profit margin influence. While previous studies have focused on local stability and codimension-1 Hopf bifurcations, this work extends the analytical framework to higher-order nonlinearities. The method of multiple scales (MMS) is employed to derive canonical forms, clarifying transition mechanisms from stable equilibria to periodic oscillations. By employing center manifold theory and normal form reduction, we analytically derive the first and second Lyapunov coefficients to characterize the Bautin (Generalized Hopf) bifurcation in the delay-free system. Furthermore, we identify the coexistence of multiple purely imaginary roots, leading to the occurrence of Double-Hopf (HH) bifurcations in the delayed model. The stability and direction of the bifurcating limit cycles are rigorously determined. Numerical simulations, including basin of attraction analysis and two-parameter bifurcation diagrams, reveal a rich dynamical landscape featuring multistability, high-order cycles, and chaotic transitions. These findings provide new insights into the mechanisms of financial instability and the emergence of complex oscillations in economic models.